Let Fq be the finite field of order q. We use this characterization of exceptional covers (finite flat morphism; of quasi-projective normal varieties) over Fq: A cover maps one-one on Fqt points for ∞-ly many t. Lenstra suggested there may be an Exceptional (cover) Tower. We construct it and its canonical limit group, permutation representation pair. The essence of the tower is in two properties of the category of exceptional covers of (Z,Fq ) of the normal variety Z:
A first characterization: You recognize exceptional covers from the first term of their extension of constants series, a purely Galois theoretic property that shows up as a Weil relation (their coefficients are the same at infinitely many terms) between the Poincare series associated to the two spaces in the cover. Such a Weil relation arises more generally if there is a chain of exceptional correspondences between two normal varieties over Fq. This gives an example of the type of Chow motive questions that exceptional covers and Davenport pairs (below) raise. Subexample: If for infinitely many t the t-th coefficient of the Poincare series of a curve over Fq is qt+1, is there a chain of exceptional correspondences from the projective line to the curve?
P(ossibly)r(educible)-exceptional covers generalize exceptional covers: Significantly different covers of a curve Z can have the same ranges on Fqt points for ∞-ly many t. We call these Davenport pairs and they produce pr-exceptional covers. An exceptional correspondence (over Fq) between two covers of Z forces them to be an example of an isovalent Davenport pair (ranges have the same multiplicities). iDPs in general produce universal relations between Poincare series. We use two problems to show how relations arise in practice.
§4 considers the crytological uses of a given exceptional tower. Restrict to covers (degree at least 2) of projective r space Pr by itself. Then, you have the natural notion of the period of the map mt for those values of t in the exceptionality set of the map. RSA uses Euler's Theorem, to compute mt as n to the qt-1 power when r=1, and the map is the n-th power. A version of this also works for twists of Chebychev polynomials.
Serre's OIT: That raises the question of considering the same question for the two major examples of exceptional covers where r=1 from Serre's O(pen) I(mage) T(heorem). There are those of prime degree from the CM part of the theorem, and those of prime degree squared from the GL2 part of the OIT. The theory of complex multiplication gives a version of Euler's Theorem to handle the computation of periods in the first case.
The GL2 case is more exciting because you can
start with one elliptic curve E over Q (given
by a non-complex multiplication j invariant) and
consider any particular prime n and the isogeny from
multiplication by n on E. Excluding a finite
set of n – exceptions to the OIT – you can descend
through Weierstrass normal form to a rational map fn:
P1→ P1 with the
following properties.